Editing Dimension (section)

===Topological spaces===

For any [[normal topological space]] {{math|''X''}}, the [[Lebesgue covering dimension]] of {{math|''X''}} is defined to be the smallest [[integer]] ''n'' for which the following holds: any [[open cover]] has an open refinement (a second open cover in which each element is a subset of an element in the first cover) such that no point is included in more than {{math|''n'' + 1}} elements. In this case dim {{math|''X'' {{=}} ''n''}}. For {{math|''X''}} a manifold, this coincides with the dimension mentioned above. If no such integer {{math|''n''}} exists, then the dimension of {{math|''X''}} is said to be infinite, and one writes dim {{math|''X'' {{=}} ∞}}. Moreover, {{math|''X''}} has dimension −1, i.e. dim {{math|''X'' {{=}} −1}} if and only if {{math|''X''}} is empty. This definition of covering dimension can be extended from the class of normal spaces to all [[Tychonoff space]]s merely by replacing the term "open" in the definition by the term "'''functionally open'''".

An [[inductive dimension]] may be defined [[Mathematical induction|inductively]] as follows. Consider a [[Isolated point|discrete set]] of points (such as a finite collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a ''new direction'', one obtains a 2-dimensional object. In general, one obtains an ({{math|''n'' + 1}})-dimensional object by dragging an {{math|''n''}}-dimensional object in a ''new'' direction. The inductive dimension of a topological space may refer to the ''small inductive dimension'' or the ''large inductive dimension'', and is based on the analogy that, in the case of metric spaces, {{nowrap|({{math|''n'' + 1}})-dimensional}} balls have {{math|''n''}}-dimensional [[boundary (topology)|boundaries]], permitting an inductive definition based on the dimension of the boundaries of open sets. Moreover, the boundary of a discrete set of points is the empty set, and therefore the empty set can be taken to have dimension −1.<ref>{{cite book |title=Dimension Theory (PMS-4), Volume 4 |first1=Witold |last1=Hurewicz |first2=Henry |last2=Wallman |publisher=[[Princeton University Press]] |year=2015 |isbn=978-1-4008-7566-5 |page=24 |url=https://books.google.com/books?id=_xTWCgAAQBAJ}} [https://books.google.com/books?id=_xTWCgAAQBAJ&pg=PA24 Extract of page 24]</ref>

Similarly, for the class of [[CW complexes]], the dimension of an object is the largest {{mvar|n}} for which the [[n-skeleton|{{mvar|n}}-skeleton]] is nontrivial. Intuitively, this can be described as follows: if the original space can be [[homotopy|continuously deformed]] into a collection of [[simplex|higher-dimensional triangles]] joined at their faces with a complicated surface, then the dimension of the object is the dimension of those triangles.{{citation needed|date=June 2018}}

{{See also|dimension of a scheme}}